Just extend the search view like every other view and add predefined filters like this: ( see "view_res_partner_filter" for example )
...
<field name="arch" type="xml">
...
<filter string="My First Value" domain="[('my_field','=', 'my_first_value')]"/>
<filter string="My Second Value" domain="[('my_field','=', 'my_second_value')]"/>
...
</field>
These predefined Filters will appear under "Filters" below the "search box" and can then be toggled by clicking ( much like your prefered behaviour with checkboxes )
If you insist on using checkboxes you will have to do a lot of work like changing the respective Qweb Templates as well as modifying the respective JS File ( see "Burmese pythis"'s Answer )
So I'd prefer this solution if time or cost matter to you.