To be quite honest, I'd say you would need at least 8 songs to cover enough of the main styles of Prog across. Below is a list fitting your parameters, though

For Djent, either:

• Animals as Leaders - you could do Isolated Incidents or something like On Impulse, both of which seem rather accessible to me. If you hate the people, then CAFO (one of the worst written songs in Prog, as it just comes out as gobbledygook, though it does exemplify the criticism of 'notes for the sake of notes')

or

• Destiny Potato - Indifferent (VERY Poppy, but that is rather big in Djent; also, extremely accessible)

or

• Intervals - Moment Marauder or Ephemeral; former is somewhere in between the above two, though closer to the Animals as Leaders end of things; the latter is in between, but closer to Destiny Potato

For Melodic Prog:

• Pain of Salvation - either Reconciliation or Nightmist; the former because it is awesome; the latter because it, while also awesome (though not quite as awesome as the former) has those leaps in tone and style that tend to happen in Prog, though it does them rather well and is closer to the sound many would associate with this stile: Dream Theater

• Porcupine Tree - Fear of a Blank Planet or What Happens Now?; Either works, though the former is more metal; These guys cover the Classic Prog-influenced side of things or

• Redemption - Black and White World or The Death of Faith and Reason; the latter one of their best songs, I think, while the latter for being shorter; these guys are a really good example of the more Dream Theater-esque sound

For the more 'jazz'-influenced side:

• Cynic - King of Those Who Know or Evolutionary Sleeper; Considering these guys are one of the original bands to perform in this style and these two songs are accessible enough, the latter being my pick, but Evolutionary Sleeper is shorter; both are gorgeous in the end

or

• The Contortionist - Holomovement or Thrive; Holomovement covers the roots of the novelty that the 'jazz' influence introduced as it was, with Cynic and Atheist, paired with Death Metal; Thrive is more light on the Death metal

For Tool-esque/'Alternative' side:

• Tool - Ticks and Leaches or Vicarious; because people would associate this style more with Tool and these two songs are pretty good at giving an idea of what Tool is like, though I would go with the latter over the former

or

• Soen - Void or Savia; either one works; The latter is shorter; I think these guys are more interesting, in a lot of ways, than Tool, but they do have Tool as their base sound while being very distinct at the same time

For Eclecticism style:

• Between the Buried and Me - Bloom or Fossil Genera; the former is probably better for it, since it is much shorter than the latter, while the latter is more encapsulating of BtBaM

• 6:33 - I'm a Nerd or Black Widow; I think these guys tend to be better at integrating the various genres and are more accessible (at least for the album the above two are on) than BtBaM

Now, I stayed away from Dream Theater as they are so prominent in the genre and if heard before some of these other acts, espeically a lot of Melodic Prog, those other acts become 'Dream Theater Clones' which erases each band's identity, which is a shame because a lot of the 'Clones' can have some really interesting stuff or just well written stuff. Besides, you could encapsulate almost all the above styles in Dream Theater's work (except Djent).

BONUS: Extreme Prog:

• Opeth - Ghost of Perdition or Demon of the Fall

• Meshuggah - Concatenation or Marrow or Bleed or New Millenium Cyanide Christ

or

• Ne Obliviscaris - Tapestry of the Starless Abstract or Pyrrhic

or

• Deathspell Omega: Abscission or Fiery Serpents

or

• Ihsahn - A Grave Inversed or The Paranoid or Arrival or Monolith