France national research institutes hinder work by banning foreign online services
First and foremost: their lab, their rules. 
As long as they is not illegal, immoral, or unethical, the lab or their funders (i.e., the government) can set standards and requirements that you must follow. If you don't like the rules, you can quit. If they catch you willfully breaking their rules, they can easily fire you for cause So I would be hesitant about circumventing their regulations if you like your job.

Second, their rules might have a rationale. Their country, their courts, their laws. European data privacy regulations are notably very strict. American companies not only operate under a looser regime -- but the companies themselves often have fictitious headquarters or locate their servers in regimes that are even looser about data privacy.  Dropbox and Google have cooperated with US courts and law enforcement to give up client data. Skype used to use clients as p2p supernodes, potentially opening up backdoors (or at minimum, using resources without permission) - this caused many agencies to forbid its use. Even leaving aside court orders and malware,  EU privacy laws are such that having something even as trivial as putting the lab's telephone directory (with personal phone numbers and addresses) on a Dropbox might mean that the data might leave the EU  -- and thus violate EU data privacy restrictions. As long as the data stays in France, it's protected by French courts and French laws. 

Third, the rules might be designed to support local industry. Just as I am only permitted to "Fly America" when using USA-government funds, it is perfectly reasonable for the French government to encourage the use of their own national technologies. If enough national lab scientists are pissed off enough about not being able to use an American software tool, they might very well come up with a superior French tool. Or so they hope.

Fourth, see rule #1: their lab, their rules. You are new to the country and to the lab. Ask your other lab members what they do. Try to not be the "ugly American" who assumes that their way is always the best way. See which rules are to be strictly obeyed and which everyone largely ignores. Learn the culture and language.