JSFiddle - React, Tailwind, and code Playground

HTML

<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.6/css/bootstrap.min.css">
<script src="https://code.jquery.com/jquery-1.12.0.min.js"></script>
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.6/js/bootstrap.min.js"></script>
<nav class="navbar navbar-fixed-top">

    <div class="navbar-default">
        <div class="container">
            <div class="navbar-header">

                <button type="button"
                        class="navbar-toggle collapsed pull-left"
                        data-toggle="collapse"
                        data-target="#megaNav"
                        aria-expanded="false"
                        aria-controls="navbar">
                    <span class="sr-only">Toggle navigation</span>
                    <span class="icon-bar"></span>
                    <span class="icon-bar"></span>
                    <span class="icon-bar"></span>
                </button>

                <a class="navbar-brand" href="#">NewCo</a>

            </div>
            <div class="additional-nav clearfix">
                <button type="button"
                        class="navbar-toggle collapsed"
                        data-toggle="collapse"
                        data-target="#siteNav"
                        aria-expanded="false"
                        aria-controls="navbar">
                    <span class="sr-only">Toggle navigation</span>
                    <span class="icon-bar"></span>
                    <span class="icon-bar"></span>
                    <span class="icon-bar"></span>
                </button>

                <button type="button"
                        class="navbar-toggle collapsed"
                        data-toggle="collapse"
                        data-target="#authNav"
                        aria-expanded="false"
                        aria-controls="navbar">
                    <span class="sr-only">Toggle navigation</span>
                    <span...

CSS

.navbar .navbar-header .navbar-toggle.pull-left {
  margin-right: 0;
  margin-left: 15px; 
}

.additional-nav {
    display: inline-block;
    float: right;
}

.navbar-header {
    display: inline-block;
}

.additional-nav .btn-group, .additional-nav .btn-group-vertical {
    position: relative;
    display: inline-block;
    vertical-align: middle;
    float: right;
    padding: 9px 10px;
}

.site-nav {
    float: right;
}