Well, yes and no. It's not like the various people who pay tuition would find it dropping by the same amount if health insurance were suspended.
The fundamental theory of unions is that the price paid for an item is a function of both supply and demand. When demand is high, the seller can charge a price higher than the cost. The question then becomes, who receives the profits?
That's not a simple question to answer, as there are a lot of inputs, but in the case of low- to moderate-skill workers, the answer is generally that the employer gets 100% of the profits. The workers are easily replaced by ones who will demand less. (In the limit case, MUCH less, and the workers are reduced to subsistence wages.) A union is a way for the workers to demand a share of the profits, by agreeing among each other not to work for the lowest offered wage.
In those circumstances, the increased wages aren't coming out of the pockets of the customers. They're coming out of the pockets of the employers. That's the point.
There are even more complex economics going on with grad students, whose "job" is being subsidized by a variety of sources, for work that is well removed from market forces. Student tuitions have been going up faster than inflation, and the grad students are competing for that extra money with a variety of campus functions (everything from fat football coach paychecks to new buildings). A grad student union is really more a representation than a true union, but it serves one of the same functions: to represent the group in the negotiation for how much they will receive of the difference between costs and monies received.