Do Eidolon's gain class levels and feats as their summoner becomes stronger or do they just have access to evolution points to get stronger?
Yes, they gain Feats.
As the summoner gains levels, their eidolon also gain feats. This can be seen on the eidolon's table for chained summoners (from Advanced Players Guide):
And also for unchained summoners (from Pathfinder Unchained):
But keep in mind that they do not gain class levels, instead, they get stronger based on his master's summoner class levels, unlocking extra hit dice, skill points, feats, evolution points and special abilities.