"Support" would be too strong a word; rather, there is nothing in this verse that necessarily argues against "eternal conscious torment." However, "with respect to this verse alone," it obviously cannot be a lone "support" for the doctrine, since the verse does not mention eternality at all (nor does it deny such); neither does it make comment at all about the state of consciousness of the one experiencing the destruction.
It should seem clear that the act of experiencing destruction could well be tormentous to an individual, that is, causing torment—not necessarily with a purposeful intent to torture, but rather merely torment as:
Any extreme pain, anguish or misery, either physical or mental
So torment may accompany how the destruction comes. If such destruction is painful, and if one has consciousness of that pain during the process, and if such a process is truly eternal, then eternal conscious torment is not ruled out. In other words, this verse alone, could support either annihilationism or eternal conscious torment.
The word απολεσαι is the aorist active infinitive of the verb ἀπόλλυμι. The aorist tense leaves undefined the duration of the action, as the author is simply making reference to the act of destruction itself, without respect to the duration (much like a "snapshot" of an event).1 It could be a moment in time, or it could be a continuously eternal process—the aorist, again, leaves that undefined.
Conclusion: No problem in the tense for maintaining eternality, yet neither does it specifically support it.
The word ἀπόλλυμι itself means (1) "to cause or experience destruction," or (2) "to fail to obtain what one expects or anticipates," or (3) "to lose someth[ing] that one already has or be separated from a normal connection."2 Either (1) or (3) would seem possible for the meaning in Mt 10:28, but given the parallelism in the verse with killing (from the verb ἀποκτείνω), (1) is the better concept, so "destruction."
But destruction or ruin does not necessarily mean ceasing to exist (i.e. annihilated), though it can mean that (such as in Mt 26:52; Jude 11). First, something can be destroyed or ruined for the purpose to which it was to serve, but still exist (such as wine-skins that burst [Mt 9:17], or food gone bad [Jn 6:27]);3 or second, something can be being destroyed, not yet having achieved a cessation of its existence, nor necessarily a finished state of ruin (such as the disciples perishing on the sea [Mt 8:25], or the prodigal son of starvation [Lk 15:17]).
Conclusion: No problem in the definition for consciousness during the process of being destroyed, nor an issue with such a process being theoretically eternal, never achieving final annihilation, yet being in a state less than "whole." Such a process can involve emotional and/or physical distress (per the disciples and the prodigal son examples), which would be a form of torment to the individual; thus no problem with there being a possibility of torment involved in destruction.
Location and Nature of such Destruction
The verse states the destruction occurs εν γεεννη, often translated "in hell," but better transliterated "in Gehenna" so as not to confuse it with Sheol or Hades, the underworld or grave, which also is often translated as "hell." Gehenna is the place of unquenched fire and perpetual worms (Mt 18:8-9; Mk 9:43-48).
While many (I think rightly) identify Gehenna as reference to the ultimate lake of fire judgment (Rev 20:14-15; 21:8), for purposes in this question, it need only be noted that the place of the destruction is labeled as Gehenna, and other revelation on that location (as noted) indicates the two key elements defining what the nature of such a destructive process entails—fire and worms.
Conclusion: Eternality is associated with two elements that cause destruction in Gehenna, and at the least, Mt 10:28 does not dispute the possibility of the destruction itself being eternal, so again, eternality is not a problem for the verse, though only tangentially present in the Gehenna reference. For an individual to be facing burning by fire and consumption by worms while still living (i.e. conscious, experiencing the process of destruction), one can hardly argue that such destruction would be a torment, so the location actually argues for the idea of torment if one is in fact conscious during the process occurring at that location. Consciousness is not explicit, though one is often conscious during the dying process (using the parallelism of the verse), and seems implied by the warnings about Gehenna's experience in other verses of Scripture.
Matthew 10:28 alone does not explicitly support eternality, torment, or consciousness, but neither does it deny those concepts. Alone, it could be used to support a case for annihilation. However, all three concepts forming the idea of "eternal conscious torment" find more explicit support with a larger contextual view defining Gehenna from Scripture. It is this definition of Gehenna that must be "plugged into" ones understanding of Mt 10:28 when it states "destroy both body and soul in Gehenna," and so one cannot isolate "this verse alone" in the discussion, for one cannot know what Gehenna means from this verse alone.4
1 Daniel B. Wallace, Greek Grammar Beyond the Basics - Exegetical Syntax of the New Testament (Zondervan Publishing House and Galaxie Software, 1999), 554-555.
2 William Arndt, Frederick W. Danker, and Walter Bauer, A Greek-English Lexicon of the New Testament and Other Early Christian Literature (Chicago: University of Chicago Press, 2000), s.v. ἀπόλλυμι.
3 ibid., 1.b.β. It is mentioned there as well a usage regarding spoiled honey in The Shepherd of Hermas, Mandate 5, i, 5 (Greek: Ἐντολὴ ε’. I.5; English translation by Lightfoot).
4 I hold the view that the resurrected bodies of the unrighteous, unclean people are just as "immortal" as that of the righteous and clean people, but not glorified like that of Christ. As such, the bodies are not only capable of being eternally burned and eaten by worms (i.e. eternally destroyed), but are in fact incapable of anything otherwise (i.e. of experiencing annihilation). Additionally, this new immortal existence makes it impossible to become "unconscious" (one does not experience shock or mental detachment because of the torment), and so is fully aware of the continuous process of destruction occurring to an ultimately indestructible body, producing a great spiritual torment of the whole soul of the individual (along with the spiritual torment of realizing that such a condition could have been avoided if one would have trusted God, being made clean and righteous).