As per US Doubles Squash rules: (emphasis added)
The two partners of a side shall serve in succession, the first
retaining his serve until his side has lost a point. On the loss of a
subsequent point the side shall be declared out and the serve
revert to the opponents. On the first serve of every game, however,
the in side shall be declared out after it has lost one point
only. The order of serving within a side shall not be changed during
the progress of a game. At the end of a game the side which has won
the game shall have the choice of serving or receiving to commence the
(b) At the beginning of each game each side shall designate one of its
players to receive service in the right hand service court and the
other to receive service in the left hand service court and throughout
the course of such game the service must be received by the players
So to answer this question:
Do players in the team have to hit the ball alternately, or can either player hit the ball?
They have to hit it alternately for serving and while receiving, the person who is assigned the right hand service court should return those hits and the other person show return hits on left hand service court. This shouldn't be changed through out the game.
Does a team have to elect a server or can either player serve or does it alternate within the team?
It is alternate within a team.
For more details on other aspects of game, you can check the above link which I mentioned.