For how long should we be avoiding sun, after using a brief treatement of Hydroquinone 4%?

This is difficult to answer without pictures. Sun protection is essential for life long maintenance, skin and aging protection. Hyperpigmentation needs a combination approach to get the most improvement with a combination of lasers, Aerolase, microneedling/prp, skin lightening, peels, and skin care. Sun protection is essential. I recommend getting a formal evaluation with a cosmetic dermatologist. Best, Dr. Emer 

I apologize in advance giving for an answer you may not like, but wearing sunscreen is always recommended. It reduces your chances of developing skin cancer and premature aging. 
To address the PIH issue specifically: your skin has already shown a propensity to hyperpigment. Once it has done this, it can often be eager to do it again. It is hard to tell ahead of time how much exposure you can get away with and not have the hyperpigmentation recur. Using sunscreen would be prudent for more than one reason.
There seems to be a second question mixed in with the first; that being about sensitivity to the sun after laser treatment. In the time following laser treatment, the skin is sensitive to the sun because the laser has (intentionally) caused injury. It is important to avoid allowing the sun to further injure the skin until it has recovered from laser treatment. This is a temporary sensitivity that occurs in the week(s) following laser treatment.
Where does this leave you? If your hyperpigmentation has cleared, you can stop using the hydroquinone. Use a broad spectrum sunscreen with an SPF of 35 to 50 every morning and reapply as necessary during the day. Wear an SPF 50 hat and get back to enjoying the outdoors. If the hyperpigmentation recurs, restart the hydroquinone and make sure you are being vigilant reapplying sunscreen every two hours.

Thank you for your question. It is difficult to give you advice without seeing pictures. However, using sun protection is essential for life long maintenance. Hyperpigmentation needs a combination approach to get an good improvement. You can use lasers, microneedling/PRP, skin lightening, chemical peels and skin care. I recommend having a consultation with an expert to discuss your concerns and to determine the best treatment plan for youl. Best of Luck!