Actually, I figured it out.  The date is using the standard datepicker - the issue is just how it is loaded in forms versus the way that it is loaded elsewhere.  After reviewing another post (https://answers.laserfiche.com/questions/166110/How-to-restrict-date-picker-to-specific-dates-of-the-month#166185) I was able to work-up a script.

Here's the walkthrough:

1. Add a date field to your form.  Select the 'yyyy-MM-dd' format (if you want to select a different format, you'll need to change a bit of the Javascript - specifically line 45).  Give it a CSS Class Name of "myLimitedDate").
2. Add a drop-down field to your form.  Give it the CSS Class Names of "blackoutDates hiddenField". 
3. Populate the blackout dates in to the drop-down by manually populating them as options on the layout page, or by doing a data lookup from a database.  You can do a combination of both if you mark the "Append choices to lookup results" box on the field.
4. Add the following CSS and Javascript to your form:

/*this is the CSS to hide any field with the hiddenField class*/
.hiddenField {display: none!important;}

//create the array of blackout dates
var blackoutDatesList = new Array();

//this function is run when the form is loaded
$(document).ready(function(){

  //set the date range of the Date field between tomorrow and 21 days in the future
  var today = new Date();
  var startDate = new Date();
  startDate.setDate(today.getDate() + 1);
  var endDate = new Date();
  endDate.setDate(today.getDate() + 21);
  var minDate = $.datepicker.formatDate('yy-mm-dd', new Date(startDate));
  var maxDate = $.datepicker.formatDate('yy-mm-dd', new Date(endDate));
  $('.myLimitedDate input').attr('min',minDate);
  $('.myLimitedDate input').attr('max',maxDate);

  //loop through each date in the blackout date list, and add it to an array
  blackoutDatesList.length = 0;
  $('.blackoutDates option').each(function() {
	blackoutDatesList.push(formatDate(new Date($(this).val())));
  });
  
  //the following four function work together to disable any dates on the picker 
  //that are weekends or are on the blackoutDatesList array
  //they also add a validation (parsley) setting for invalid date entries
  function isValidDate(n) {
    var t = n.getDay();
    var d = n.getDate();
	var string = jQuery.datepicker.formatDate('yy-mm-dd', n);
    return (t!=6 && t!=0 && blackoutDatesList.indexOf(string) == -1);
  }
  function checkDate(n) {
    return[isValidDate(n),""];
  }
  function checkField(input, inst) {
    if ($(input).closest('li').hasClass('myLimitedDate')) {
      $(input).datepicker("option", {beforeShowDay: checkDate});
    }
  }
  $.datepicker.setDefaults( {beforeShow: checkField} );
  window.Parsley.addValidator('limiteddate', {
    validateString: function(value) {
      //IMPORTANT! this date format needs to match the format you set for the field on the layout page
      return isValidDate(moment(value, "YYYY-MM-DD").toDate()); 
    },
    messages: {
      en: 'Not a valid date.'
    }
  });
  $('.myLimitedDate input').attr('data-parsley-limiteddate','');
});

//this function is run when the lookups are completed.
$(document).on("onloadlookupfinished", function () {
  //loop through each date in the blackout date list, and add it to an array
  blackoutDatesList.length = 0;
  $('.blackoutDates option').each(function() {
	blackoutDatesList.push(formatDate(new Date($(this).val())));
  });
});

//function to return any date formatted as yyyy-mm-dd
function formatDate(date) {
  var d = new Date(date),
    month = '' + (d.getMonth() + 1),
    day = '' + (d.getDate() +1),
    year = d.getFullYear();
  if (month.length < 2) 
    month = '0' + month;
  if (day.length < 2) 
    day = '0' + day;
  return [year, month, day].join('-');
}

The script is allowing any date between 1 and 21 days from now (that range is set on rows 10 and 12 of the Javascript).  Then it excludes any date that a Saturday or Sunday or is on the list of blackout dates.

In my case, I listed these dates as choices within the drop-down menu:

03/23/2020
03/24/2020
03/25/2020

And then these additional dates are being added via database lookup:

2020-03-31
2020-04-01
2020-04-07
2020-04-08
2020-04-09

The end result, is that with today being Friday, March 20, 2020, we see these active dates:

3/21 is the start of the range, 3/21-3/22 is a weekend, 3/23-3/25 are on the blackout date list, 3/28-3/29 is a weekend, and 3/31 is on the blackout date list.

4/1 is on the blackout date list, 4/4-4/5 is a weekend, 4/7-4/9 are on the blackout date list, and 4/10 is the last date allowed in the range.

***EDIT - changed line 67 of the script to match @████████ 's suggestion.***