Based on what you said, I'm pretty sure you've ruined the water filter. If you're sick after using the filter, then it's not doing its job anymore. Now, let's look at some facts:
I hiked near freezing point, but filter was in backpack and tent during night and I don't think it froze.>
"I don't think it froze". It certainly did. Water conducts cold 25x faster than air. So the filter may have frozen even though you "think" it didn't. Freezing water expands, so it destroys the hollow fibers inside it. You must keep the filter inside your jacket at all times when it gets cold. The best is to use a ziplock bag so you can have it inside your jacket during the day, and inside your sleeping bag at times. "Inside the tent and backpack" doesn't cut it. The filter you're using is a Sawyer (The syringe). It's a very good brand, but like everything else it has to be taken care of. Those filters are made of hollow fibers and if frozen, they will not do the job anymore.
Also, you used the syringe, but did you use CLEAN water and did a REVERSE flush? (asking just to make sure". And if you used clean water, did you use tap water? If so, it's most likely treated water with all kinds of stuff that will clog any filter over time. The best is to use water you just filtered or
bottled water (best is DE-mineralized water).
I'd suggest you buy a new Sawyer ASAP and make sure to keep it away from the cold next time.