Only when the caster is below 5th level

The errata for the PHB has clarified the restriction further from the original printing:

To be eligible for Twinned Spell, a spell must be incapable of targeting more than one creature at the spell’s current level.

By default, eldritch blast does not have a range of self and is capable of targeting only one creature. It is thus eligible to be twinned.

However, the spell becomes capable of targeting more creatures once the caster reaches level 5:

The spell creates more than one beam when you reach higher levels: two beams at 5th level, three beams at 11th level, and four beams at 17th level. You can direct the beams at the same target or at different ones. Make a separate attack roll for each beam.

Thus, once the caster reaches 5th level and above, they can no longer twin eldritch blast.

Note that eldritch blast (and all other similar cantrips) scale with character level not class level as confirmed (unofficially) by Jeremy Crawford on Twitter. (See Do Cantrips use your character level or class level? for more discussion about that)

Yes, as long as the cantrip has a casting time of 1 Action

Most cantrips require an action to cast. As such, you can certainly quicken them. So any 1 Action casting time cantrip will be usable, such as Eldritch Blast in this question.

However, PHB p. 202 states:

A spell cast with a bonus action is especially swift. You must use a bonus action on your turn to cast the spell, provided that you haven't already taken a bonus action this turn. You can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.

Thankfully, if you are planning on casting two cantrips per turn, you are more than able to do this effectively, as they are not restricted by the same restriction as regular spells.