Do shields have hit points?
How many?
Directly themselves as in some other games, no.
The flux itself does roughly the same job though. The more flux a ship can 'hold', the longer it can keep its shields up.
Effectively, a shields hit points is the ships flux pool.
Adding capacitors, a hullmod or taking one particular skill can increase this.
Shields also have an efficiency rating which will be a decimal value somewhere near 1.
Where 1 means every 1 point of damage gets turned into 1 point of flux.
Lower numbers mean more efficiency and higher means less.
0.7 means every 1 point of damage gets turned into 0.7 points of flux.
Adding a hullmod or taking a specific skill can increase the efficiency.
Is there a way to check?
For your own fleet you can access the refit screen, or use the information infocard on the fleet buy/sell to pull data for each ship individually.
Improvements over the baseline for that hull type will be appended with a green number telling you how much better it is.
Conversely, areas below baseline will be appended with red numbers.
For both your own fleet and enemies in battle, if you hold your cursor over a ship a mini-display will pop up just above them and show you two bars.
The bottom bar is hull integrity. Basically the health bar, when it hits zero the ship is disabled.
The top bar is the current flux level. Watching this will allow you to press an attack while an enemy can't fight back effectively.
When this bar is full a ship will try to pull away from battle to vent the flux, and will be forced to fire weapons sparingly and lower shields.
What happens after they get depleted?
This:
What I'm noticing is enemy ships dropping their shields after a while and raising them up right after, giving a small window to shoot something else at them.
Or they don't drop the shields fast enough and overload.