Hi BP~
Here's a suggestion that might work: Gather some snow from outside and bring it INDOORS. Place the snow in a small kiddie pool or something. (I got snow and placed it in my kids' plastic toddler bathtub. I placed a blanket underneath the little tub. They would sit near the edge of the tub and scoop snow into containers and explore snow. (Kind of like playing in the sand, but with snow instead. Yay!! Let's build snowcastles!) My kids weren't afraid of snow but sometimes I just brought some inside and they could play and not be cold while I watched them, I could clean the kitchen. A little food dye in a water spray bottle is always fun too! They loved spraying two different colors on the snow and would investigate the change of snow color.
He still may want to wear mittens inside because his hands might get cold if he continues to touch the snow with bare hands. Or if you want to go along with his idea that the snow is dirty...... then give him a water spray bottle filled with colored water and tell him that he can go have fun and CLEAN the snow!! My kids were good about keeping the snow in the plastic tub, but if he does have a problem with this, give him a warning and tell him if he wants to play with it, it HAS to STAY in the pool/tub. If snow gets tossed out again, then the fun of bringing it inside is over for the day. If he enjoys my idea, maybe after awhile he'll venture into playing outdoors in the snow.
Hope this helps!