Can Unity play Mechanical Golems when she gets to play a card outside of her turn?
Yes, she can play a mechanical golem outside of her play phase as if it was any other card. All that card is saying is you can't play it during the play phase and it does not mention not being able to play it like any other card.


  This card cannot be played during your play phase.


Take this card for instance


  Scrap Metal - "At the end of your turn, either take 1 Mechanical Golem from play back into your hand, or destroy this card.
  At the start of your turn, you may play a card. If you do, destroy this card."


What that card lets you do after you take a mechanical golem from play in into your hand you can then play it during the start phase of your next turn allowing it to  return the play with full health. What this means is that you can play a mechanical golem during your turn as long as it is not the play phase. The start of turn, user a power, draw a card and end of turn steps (or during any other player/environment/villain turn) are perfectly legal to play a mechanical golem as long as you gain the ability to play a card during those steps.