I think if you use type="radio" instead (so that only one can be selected) then it will work. But there's a problem with the $emailchecked values since they are either selected="selected" for the selected choice or null, and a radio button needs checked="checked" instead. But something like this might work:
Code:
<input type="radio" name="emailupdate" value="0" tabindex="1" <if condition="$emailchecked[0]">checked="checked"</if> />
<input type="radio" name="emailupdate" value="1" tabindex="1" <if condition="$emailchecked[1]">checked="checked"</if> />
<input type="radio" name="emailupdate" value="2" tabindex="1" <if condition="$emailchecked[2]">checked="checked"</if> />
Did you mean to leave out the "no email notification" choice?