I'm still not sure what was causing the generated UniqueIDs in the MasterPage to be different in Page_Load than when rendered to the page. However, I was able to get around the issue by storing the UniqueIDs of these fields in hidden fields. I was then able to access the values directly in the Request.Form collection.
In other words, I did this:
In the MasterPage -
<input type="text" runat="server" id="txtUserName"/>
<input type="text" runat="server" id="txtPassword"/>
<input type="hidden" id="txtUserNameUID" value="<%=txtUserName.UniqueID%>"/>
<input type="hidden" id="txtPasswordUID" value="<%=txtPassword.UniqueID%>"/>
During Page_Load of the child page -
string username = Request.Form[Request.Form["txtUserNameUID"]];
string password = Request.Form[Request.Form["txtPasswordUID"]];
Hope this helps anyone else struggling with UniqueID weirdness in ASP.NET!