How can I translate checkboxes on a frontend user registration form to lightswitch custom fields?
When the checkbox get submitted, it get's transferred with whatever value is set with the value attribute.

Thus, your need to set the value to 1. Your code should be more like this:

&lt;div class="checkbox"&gt;
  &lt;label&gt;
    &lt;input type="checkbox" id="memberShowEmailLink" name="fields[memberShowEmailLink]" value="1" {% if account is defined and account.memberShowEmailLink %} checked="checked"{% endif %}&gt;
    Show a link to contact me via email in the member directory?
  &lt;/label&gt;
&lt;/div&gt;