H1-b簽證已經六年無法續簽了，公司還可以幫辦綠卡嗎？
-Yes.
若辦NIW是否就和簽證過期無關了？
-Correct.
You need to have a legal status to stay in US such as F1. Otherwise, you have to leave US and then, apply for the immigrant visa to enter US when your priority date is current in the future either through the company sponsor sponsor or your NIW.
Another choice is that you leave US and stay outside US for one year and then, apply for H-1 visa to return to US if the US company sponsor your H-1, but you have to be selected randomly by USCIS in the H-1 visa lottery unless you work for a college/university or non-profit institution.
