The claim is true but possibly only applicable to a very small range of situations.
In 1953, as part of Operation Upshot Knothole, tests were carried out on miniature buildings to test the effects of extreme heat on the buildings. They did find that wooden buildings surrounded by dry combustible material were more likely to catch fire than clean white-painted buildings lacking any large quantities of loose combustible material.
Project 8.11a, Incendiary Effects on Building and Interior
Kindling Fuels, was fielded by the Forest Products Laboratory,
Forest Service, Department of Agriculture. The project was
designed to study the vulnerability of urban structures to
primary fires produced by nuclear detonations. The study focused
on materials that were either part of a building or were found
within a building. Before Shot ENCORE, personnel placed
furniture in two block houses and materials outside of three
small frame houses specially constructed for the project. For
both Shots ENCORE and GRABLE, personnel placed wooden racks with
materials such as newspapers, weeds, and rags at various
distances from ground zero. They returned after each shot to
inspect damage (52)
From Report DNA 6014F - OPERATION UPSHOT-KNOTHOLE 1953
The Effects of Nuclear Weapons
This was allegedly the title of a 1957 book published by the DoD.
only 12 cal/cm2 was needed at the 1953 Encore nuclear test to ignite houses made of rotted wood or surrounded by a trash filled yard and wooden fence (the whitewashed wooden house with a clear yard survived). Two wooden houses were also constructed for that test, exposing 4 x 6 foot windows with a line-of-sight exposure to ground zero. Both were subjected to the same 17 cal/cm2 thermal flash from the Encore nuclear test, and the one full of inflammables ignited with immediate flash-over to the entire room, while the one with modern fire-resistant furnishings survived with just minor smouldering which was extinguished by the recovery party when they entered the house an hour after the test.
So yes, if you paint your wooden house white and remove nearby combustible materials it is less likely to catch fire if there is a nuclear strike on a nearby city.
National Paint ... Association
The "National Clean Up-Paint Up-Fix Up Bureau" were an arm of the "National Paint, Varnish, and Lacquer Association". They used information from the nuclear tests to promote their products.
It is notable that they avoided any analysis of how narrow the range of distances from epicentre might be where such differences in household tidiness would affect the outcome. Any closer, and the houses would be knocked over by blast. Much further and the dried grass around the untidy houses might fail to ignite.