Why will BoxLayout not allow me to change the width of a JButton but let me change the height?

You may not want Box.createGlue(), which "grows as necessary to absorb any extra space in its container." Instead, use Box.createVerticalStrut() between the buttons, as shown below and in the ControlPanel of this simulation.

example.setLayout(new BoxLayout(example, BoxLayout.Y_AXIS));
example.add(findNext);
Box.createVerticalStrut(10);
example.add(cancel);

Addendum:

adding in setMaximumSize() made it work.

This is the expected behavior for components having identical maximum widths in a vertical BoxLayout, as described in Box Layout Features. The preferred width of the container becomes that of the (equally wide) children, and the X alignment becomes irrelevant.

example.setLayout(new BoxLayout(this, BoxLayout.Y_AXIS));
JButton findNext = new JButton("Find Next");
JButton cancel = new JButton("Cancel");
Dimension d = findNext.getMaximumSize();
cancel.setMaximumSize(new Dimension(d));
example.add(findNext);
example.add(cancel);

As mentioned in the comments on the question, you were able to fix it by switching to setMaximumSize(). However, as you noted, setPreferredSize() doesn't work. So, what's up with that?

With many things Swing, the properties used to determine the actual component size when using the BoxLayout are somewhat random (in my opinion). When determining how to render the components, Swing calls layoutComponent() on the layout manager, which is figures out where to position everything.

BoxLayout's implementation of layoutComponent() involves a call to a method that creates SizeRequirements objects for the width and height of each of the components you add to the JPanel, based on their getMinimum/Preferred/MaximumSize() methods.

Later, it calls SizeRequirements.calculateAlignedPositions() for determining the correct width values for each component, because your orientation is BoxLayout.Y_AXIS (The heights are calculated using a different method). Taking snippets from the source, the relevant implementation of this method is as follows:

for (int i = 0; i < children.length; i++) {
    SizeRequirements req = children[i];
    //...
    int maxAscent = (int)(req.maximum * alignment);
    int maxDescent = req.maximum - maxAscent;
    //...
    int descent = Math.min(totalDescent, maxDescent);
    //...
    spans[i] = (int)Math.min((long) ascent + (long)descent, Integer.MAX_VALUE);
}

Note that totalDescent is the available width, so descent is always set to maxDescent, which is based on SizeRequirements.maximum, which was taken from JButton.getMaximumSize(). The value of spans[i] is then used later in a call to JButton.setBounds() as the width. As you'll note, getPreferredSize() was never involved here, which is why setting it has no impact in this case.

Usually if want to ensure a size of the component in Swing you need to call setMinimumSize(), setMaximumSize(), and SetPrefferedSize() with the same value.

button.setMaximumSize(getMaximumSize());

If you put your buttons in a GridLayout panel they will be the same width.