Hello @vitalik,
Please once go to Admin Panel > WooCommerce > Settings > Accounts > My Account Page and select the page in this filed . Without assigning the page in this field will lead to 404 page when clicked on register button.
Also please make sure you have checked the field to enable the Enable Registration, Login, Account Creation.
Refer to below screen shot for more info.
http://image.prntscr.com/image/13e58b0fec4c41e59ce63162e3b241ff.png
Hope this helps.
For any confusion further, please let us know.
Best Regards!!