I don't but all I have is an antenna in the attic so I don't think it has a great chance of being struck by lightning. The purpose of those is to prevent a surge coming from the cable. What is your co-ax hooked to (e.g. cable, satellite, etc.)?
I had it running through the surge protector (Monster) but the picture wasn't quite as clean as before so I removed it.
I talked to the cable guy recently whho said that my boxes (Scientific Atlanta 8300HD) had a built in surge protector. Is it as good? I dunno, and I hope to never find out.
One thing the coax connector on a surge protector does is tie the "shield" of the CATV coax to your power ground (the third prong on the power plug).
Sometimes - this solves hum that comes in on your CATV line (The CATV picks up 60 hz by running in parallel with power wires.)
But - it is a break in the signal path. If you were feeding a weak analog CATV signal through it - I would try to avoid the extra break. If the coax is bringing in a digital signal - it's a lot less likely to cause problems running through a power strip.