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According to the Big Squiggoth rules, 15 models may be transported by it. Does this mean they can shoot while in the Squiggoth?

That is a great question, that i don't have the relevant rules to answer properly.

Does it have any fire points or count as an open-topped vehicle for the transport rules?

It has to have some rules on embarkation/disembarking and how to treat carried passengers. These rules should tell you if passengers can fire.

The rules say it counts as an open topped transport.

Then yes they may all fire