JSFiddle - React, Tailwind, and code Playground

HTML

<link rel="stylesheet" href="//maxcdn.bootstrapcdn.com/bootstrap/3.2.0/css/bootstrap.min.css">
<nav class="navbar navbar-default" role="navigation">
    <div class="navbar-header">
        <button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-collapse">
            <span class="icon-bar"></span>
            <span class="icon-bar"></span>
            <span class="icon-bar"></span>
        </button>    
    </div>
    <a class="navbar-brand navbar-center" href="#">
        <img src="http://placehold.it/100x40" alt="Logo Here">
    </a>
    <div class="navbar-collapse collapse">
        <ul class="nav navbar-nav navbar-left">
            <li><a href="#itemonelink">Item One</a></li>
            <li><a href="#itemtwolink">Item Two</a></li>
        </ul>
        <ul class="nav navbar-nav navbar-right">
            <li><a href="#signuplink">Signup</a></li>
            <li><a href="#loginlink">Login</a></li>
        </ul>
    </div>
</nav>

CSS

.navbar-brand {
    float: none;
    padding: 5px;
}
.navbar-center {
    position: absolute;
    width: 100%;
    left: 0;
    top: 0;
    text-align: center;
    margin: auto;
    height:100%;
}