Usually first thing to go on these engines is the regulator. It's a little black box with cooling fins. Unregulated, it should produce about 15 volts. If there is nothing, my guess the stator is bad.
[ame]http://www.amazon.com/31620-ZG5-033-Genuine-Honda-Regulator-Rectifier/dp/B00S7QN0JM/ref=pd_sim_sbs_86_1?ie=UTF8&refRID=1JQA3SF1GBF7N6GHD0PA&dpID=41s4mphPVdL&dpSrc=sims&preST=_AC_UL160_SR160%2C160_[/ame]
[ame]http://www.amazon.com/Honda-GX610-GX620-CHARGING-STATOR/dp/B00FOUSA7Q/ref=sr_1_1?s=lawn-garden&ie=UTF8&qid=1442942831&sr=1-1&keywords=gx620+stator[/ame]
[ame]http://www.amazon.com/31620-ZG5-033-Genuine-Honda-Regulator-Rectifier/dp/B00S7QN0JM/ref=pd_sim_sbs_86_1?ie=UTF8&refRID=1JQA3SF1GBF7N6GHD0PA&dpID=41s4mphPVdL&dpSrc=sims&preST=_AC_UL160_SR160%2C160_[/ame]
[ame]http://www.amazon.com/Honda-GX610-GX620-CHARGING-STATOR/dp/B00FOUSA7Q/ref=sr_1_1?s=lawn-garden&ie=UTF8&qid=1442942831&sr=1-1&keywords=gx620+stator[/ame]
