The Problem
In some browsers the <button> element doesn't accept changes to its display value, beyond switching between block and inline-block. This means that a <button> element cannot be a flex or grid container, or a <table>, either.
In addition to <button> elements, you may find this constraint applying to <fieldset> and <legend> elements, as well.
See the bug reports below for more details.
Note: Although they cannot be flex containers, <button> elements can be flex items.
The Solution
There is a simple and easy cross-browser workaround to this problem:
Wrap the content of the button in a span, and make the span the flex container.
Adjusted HTML (wrapped button content in a span)
<div>
    <button>
        <span><!-- using a div also works but is not valid HTML -->
            <span>Test</span>
            <span>Test</span>
        </span>
    </button>
    <p>
        <span>Test</span>
        <span>Test</span>
    </p>
</div>
Adjusted CSS (targeted span)
button > span, p {
    display: flex;
    flex-direction: row;
    justify-content: center;
}
Revised Demo
References / Bug Reports
Flexbox on a <button> blockifies the contents but doesn't establish a flex formatting context
User (Oriol Brufau): The children of the <button> are blockified, as dictates the flexbox spec. However, the <button> seems to establish a block formatting context instead of a flex one.
User (Daniel Holbert): That is effectively what the HTML spec requires. Several HTML container-elements are "special" and effectively ignore their CSS display value in Gecko [aside from whether it's inline-level vs. block-level]. <button> is one of these. <fieldset> & <legend> are as well.
Add support for display:flex/grid and columnset layout inside <button> elements
User (Daniel Holbert): 
<button> is not implementable (by browsers) in pure CSS, so they are a bit of a black box, from the perspective of CSS. This means that
  they don't necessarily react in the same way that e.g. a <div>
  would.
This isn't specific to flexbox -- e.g. we don't render scrollbars if you put overflow:scroll on a button, and we don't render it as a
  table if you put display:table on it.
Stepping back even further, this isn't specific to <button>. Consider <fieldset> and <table> which also have special rendering
  behavior.
And old-timey HTML elements like <button> and <table> and <fieldset> simply do not support custom display values, other than
  for the purposes of answering the very high-level question of "is this
  element block-level or inline-level", for flowing other content around
  the element.
Also see: