I found some information on Chizhevsky's work at a Cycles Research Institute site, and downloaded his Physical Factors of the Historical Process. I wasn't impressed.
He picked out historical events and attempted to match them with sunspot cycles. The means by which he picked them out isn't obvious, and the assignment of exact dates is frequently arbitrary. Since this is matched against time periods of a few years each (the usual 11-year cycle is divided into four sections), there are certainly possibilities of cherry-picking here, and no information on how that was guarded against, and if it was. There is a list of cholera epidemics, and no mention of other diseases. The cycles for most of his period of study are measured nine to a century, with only moderate attempts to account for the actual cycles.
The attempt to break WWI down by sunspot cycles seems forced. Having read several books about it, I wouldn't have guessed 1915 as the period of greatest violence, although he claims that as a peak of violence. There was a lot of fighting against Russia that year, although less than in 1916's Brusilov offensive, and not a whole lot in France and Belgium, particularly compared to 1916's Verdun and Somme battles. He attributes the 1917 Russian Revolution to another outburst of sunspots, but overall the fighting had abated some since 1916, with the Russian and French armies seriously demoralized and incapable of large offensives.
The article that you linked was sensationalistic, and tied the maxima of 1990 to the Kuwait wars and 2001 to one (admittedly large) act of terrorism and two invasions (one in 2003), ignoring other cycle points. Counting back from 1990 by elevens, we get 1979, which was fairly quiet, 1968, when the Vietnam War was about to wind down but still raging (like the Tet Offensive), 1957, a fairly quiet year over most of the globe, and 1946, when very little of military consequence happened, as the world tried to recover from WWII.
It seems to me that any major disturbances should show up more clearly in crime rates, but I saw nothing about crime rates in a quick look at the CRI site, and I've never heard of an eleven-year cycle there.
In short, I have failed to see appropriate evidence for the claim.