Should more Terran players be using bunkers instead of other buildings to wall off?
All decent Terran players use a bunker to defend their natural against Protoss.
Is there a better tactic I could have used to apply pressure early game?
This is hard to answer as you haven't been very specific about what time this has all happened. If you add more detail or even a replay, I'll be more than happy to go through it and change my answer to suit the situation better.
First of all, you're not going to beat bunkers being repaired in a head on fight. Some things you can do are:
Run past the bunkers
This is good for a mainly stalker composition, more easily executed if they don't have marauder slow. You just run right past the bunkers and into the main or even natural. You can kill workers quite easily and if he leaves his bunker you can probably beat his army pretty easily, unless he pulls workers and surrounds you, in which case it'll be bad for you so watch for this.
If you bring a sentry or two as well you can run into his main and forcefield his natural ramp, to prevent him from coming up, you can cause lots of damage to his workers this way, however sentries are weak and slow, it's likely that they will die when attempting to run past.
The problem with this is that your units are nearly guaranteed to die, but it's likely that they will pay for themselves, of course it all depends on how much firepower they have for you to run past, if it's too much, you'll lose too many trying to get into the base making this a bad choice.
Another tip is to lead the assault with a zealot, zealots are tougher and can soak up more damage allowing the rest of your stalkers to get up past the bunkers.
Focus the repairing workers
This is best if you want to play it safe. Just poke up with a zealot or two and some stalkers, use the stalkers to focus fire the workers that come to repair.
You can go back down after your zealots die, wait for more reinforcements, or just for your shields to regenerate then go up a few more times and do the same thing.
Don't stay too long, marauder slow will guarantee you lose some stalkers.
Watch out for the enemy unloading and running down the ramp as you go up!
This is a lot more of an investment into this attack, and will likely win or lose you the game. This is where you bring 4-10 sentries and then just smash into the front with a decent army.
Don't get too many stalkers in this case, zealots not only are cheaper and tougher but have more dps, especially when they can't run away.
Which is what all the sentries are for. When he pulls his scvs to repair, forcefield the back half or so of the bunkers to stop his scvs from getting in range of the bunker to repair it (your zealots will be blocking the front half or so) without the repairing, you can probably bust the bunkers down pretty quick. Also don't forget a guardian shield, it lowers marine dps drastically.
Use any remaining forcefields to stop his forces from retreating. Zealots dps is high but only if they can actually attack. if you forcefield behind the enemy forces, they can't run away and the zealots will do tonnes of damage.
Make sure you're constantly warping in during this. Keep reinforcing and depending on how the fight goes you can usually kill off his natural. If you see you are going to bust through initially, prevent him from retreating to his main with a forcefield on the ramp, this way you'll be able to go up into his main and win the game. If he retreats enough units into his main it's unlikely you'll win right then. He'll float his natural up and survive, but you'll have done a lot of damage so you can retreat at this stage and focus on the next stage of the game.