I fly an Eagle and use gimbaled pulse lasers and missiles. I've spent time in the federal distress signal area as recently as an hour ago. No idea whether you were a victim, but if you were it is simply because you were flagged as an enemy.
An hour prior I was minding my own business when out of nowhere I had a Sidewinder on my tail. I tried to outmaneuver, and succeeded, but then I made the mistake of turning into him and he rammed me, and I blew up. It was quite epic. The shields/hull on the Eagle seem paper thin compared to a Sidewinder. Not sure if he blew up.
The Eagle is great for quickly swooping in and out at speed, but head-on with a Sidewinder can often spell death. If I equip 3 pulse lasers they overheat almost instantly - so 3 lasers are largely unusable. With missiles, it takes time to get a lock. So in 8 of 10 situations I'm stuck with 2 working lasers - just like a Sidewinder, and certainly what feels like a thinner shield/hull, though I have speed and maneuverability on my side.
On top of that - for some inexplicable reason - my lasers like to stop working frequently for no particular reason (but only on the Eagle, not in a Sidewinder - even with full power in Weapons, and full "power" in available lasers). I get much longer, more reliable and sustained laser bursts from a Sidewinder.
I picked the Eagle for speed and occasional use of missiles. But it's not a massively superior ship. In fact, generally speaking, he who sees the enemy first - and lands the first shot - tends to win the battle, whether it be a Sidewinder attacking an Eagle or vice versa.
Short of the long: I've been killed by Sidewinders in an Eagle. And I've destroyed Eagles in a Sidewinder. Timing, positioning, skill and a bit of luck outweigh ships/equipment when it comes to these 2 ships.