Do you have to use hydroquinone before and after non-ablative laser resurfacing to get results?

It certainly does sound as if you are having a reaction to the hydroquinone and I would recommend that you stop it.  In my experience about 5-10% of patients don't tolerate hydroquinone. If your physician feels strongly about pretreating then the are other hydroquinone-free alternatives such as Skin Medica's Lytera.  The only patients that I recommend pretreating with hydroquinone or Lytera are darker skin types or patients with significant melasma. In short,  a non-ablative laser treatment is still an option for you but you need to consult with your physician to come up with a agreed upon pre and post treatment skin care regimen that will give the best result.  Good luck. 

You should stop using the hydroquinone.  There are other skin lightening products such as Lytera that you might be able to tolerate.  When you have a laser resurfacing, you want to protect your investment so using some retinol or skin lightening product afterwards is to your advantage.