find select option by text

You are using Attribute Equals selector which selects elements that have the specified attribute with a value exactly equal to a certain value, option elements don't have text attributes, you can use :contains selector instead, try this:

Select all elements that contain the specified text.

$(function(){
    $('#my_button').click(function(){
        var unitName = "Unit2";
        $('.assUnit').find('option:contains('+unitName+')').remove();
    });
});

FIDDLE

If you want to select the element that has only certain value you can use the filter method:

$(function(){
    $('#my_button').click(function(){
        var unitName = "Unit2";
        $('.assUnit option').filter(function() {
             return $(this).text() === unitName
        }).remove();
    });
});

FIDDLE

You will probably have more luck with this:

$('.assUnit').find('option:contains('+unitName+')').remove();

See also: :contains() selector