[Wolfram]

I use Dunlop rackets, but most of the brands have decent ones. Basically, just chose from the top price bracket of the brand and try before you buy. I've used Dunlop, Wilson, Gray's and Black Knight, and the only ones I didn't like were Black Knight (I had issues with their durability).

And buy new grips and change them often.

You can use any type of light, flat-soledm indoor shoe for squash. Don't use basketball/running/cross-training shoes because the sole is to thick and it increases risk of ankle injury.
I really like Adidas Stabil. If you play more than twice a week you should by new shoes every year.

Squash plays havoc with the skin on the toes and soles of your feet so I'd recommend getting good new sports-socks to avoid blistering and chafing.

Headbands and wristbands are a matter of preference. I use a rolled up bandanna to keep sweat out of my eyes. My hands are usually pretty dry so I don't need wristbands.

If you have any other squash related questions don't hesitate to ask.