Though heavy rain and snow or very heavy cloud cover can certainly affect the service, it shouldn't go out every time there is precipitation. It's possible that there is a dish alignment issue that's making the signal especially susceptible to "rain fade".
If you don't already, please be sure to leave your modem plugged in so that a rep will have the ability to run remote diagnostics on your HughesNet equipment if they deem it necessary. The reps are on M-F from approximately 8AM to 5PM EST. It can take them a day or so to reply, but they've been pretty quick as of late.
In the meantime, when you go to this page, what is the Satellite Receive Signal Strength that's showing in the WAN Info box on the bottom left?
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