Does the .shtml extension hurt your rankings in Google? More
specifically, is it "worse" than .html?

URLs : Apart from the Domain Name, directory names and filenames are
very important to Google.  These won't help as much as a keyword rich
Domain Name.  But it does help.

Here's how Google sees files on a server... the most important page is
the homepage at the root.  Next are the directory and subdirectory
homepages (up to 2 deep).  Next are the files on the root named
something other than what is thrown up by default.

Even your file name extension has importance.  An older study also has
.shtml rank highest, then .shtm, html and htm in that order for
straight web pages.  .php actually ranks best for dynamic pages, then
.asp.

So here is a list of possibilities in the order that they might rank best:

server.com/index.php
server.com/default.php
server.com/home.php
server.com/index.asp
server.com/default.asp
server.com/home.asp
server.com/index.shtml
server.com/default.shtml
server.com/home.shtml
server.com/index.shtm
server.com/default.shtm
server.com/home.shtm
server.com/index.html
server.com/default.html
server.com/home.html
server.com/index.htm
server.com/default.htm
server.com/home.htm
server.com/keyword/index.php
server.com/keyword/default.php
server.com/keyword/home.php
server.com/keyword/index.shtml
server.com/keyword/default.shtml
server.com/keyword/home.shtml
server.com/keyword/index.shtm
server.com/keyword/default.shtm
server.com/keyword/home.shtm
server.com/keyword/index.html
server.com/keyword/default.html
server.com/keyword/home.html
server.com/keyword/index.htm
server.com/keyword/default.htm
server.com/keyword/home.htm
server.com/keyword/keyword2/index.php
server.com/keyword/keyword2/default.php
server.com/keyword/keyword2/home.php
server.com/keyword/keyword2/index.shtml
server.com/keyword/keyword2/default.shtml
server.com/keyword/keyword2/home.shtml
server.com/keyword/keyword2/index.shtm
server.com/keyword/keyword2/default.shtm
server.com/keyword/keyword2/home.shtm
server.com/keyword/keyword2/index.html
server.com/keyword/keyword2/default.html
server.com/keyword/keyword2/home.html
server.com/keyword/keyword2/index.htm
server.com/keyword/keyword2/default.htm
server.com/keyword/keyword2/home.htm
server.com/keyword-keywords.php
server.com/keyword-keywords.shtml
server.com/keyword-keywords.shtm
server.com/keyword-keywords.html
server.com/keyword-keywords.htm

Filenames are literally almost anything including graphics.  Name your
graphic images and Flash elements with keyword rich names.  Use the
alt tag or title tag to accentuate the keyword(s) by repetition (but
only once per title/alt tag unless it is also in your business name).