Thanks to an interesting post titled "Responsive ASP.NET Menu Control With Twitter Bootstrap", I was able to update the Bootstrap with master pages example with asp.net menu controls instead of being hard coded.
For the menu control, you basically add the below code in lace of the div with "nav-collapse collapse". The menu in this code is based off the Bootstrap Example "Basic marketing site".
Here is a link to the raw code in GitHub. It starts at line 43.
The one downside to the menu control is that the control will inject inline styles. Also, to display the search form in the menu bar, use the code from the bootstrap documentation, then change the <form> to a <span>.
<form class="navbar-search pull-left"> <input type="text" class="search-query" placeholder="Search"> </form>
<span class="navbar-search pull-left"> <input type="text" placeholder="Search"> </span>