What is the best solution for broken flagstone?

That is a question only a professional can answer after inspecting your house.  It may be a simple point and reset job, or it could be more complicated due to the base of the wall sheathing having developed leaks and now rot behind the flagstones.  Call a mason in the area that has been in business for more than 10 years and get an opinion from someone who knows this application, and then get a quote from a long time vinyl siding company (Check that they have their VSI Certification (Vinyl Siding Institute) and/or Manufacture Certified Ex. Certainteed has 5-Star Installers).  I say recommend long-standing companies because these people have the expertise that only time can provide. An experienced company will be better able to determine what is "behind" that surface problem, and you want this fix to last. So call the pro. (National Association of the Remodeling Industry  http://www.nari.org/)