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がい(甲斐) is a word you can stick on the end of a verb that makes it meaning something like the benefit of taking that action, the good outcome that results from that action.

One way I've often seen this used is 生きがい. 生きがい is what makes life worth living, 生きがいがある is having something to live for. Another common one is やりがい.

So 搾りがいがある means there's some kind of benefit to squeezing something.

Sorry I suck at explaining, I hope you can understand with the examples.