The final squash tournament of the 1983 winter season starts Friday evening at the Washington Squash Racquets Club. Squash fans are invited to catch the action, free, at the Booeymonger/Amstel Open Championship where top-ranked amateur players compete for trophies and national ranking. First-round matches start Friday at 6 and go until 11; second-round, quarterfinal and consolation matches are Saturday from 9 to 6; and semifinals and finals are Sunday from 10 to 5. The club is at 1120 20th Street NW between L and M. Call 659-9570. And if watching all that action makes you itchy for exercise, then Sunday's fourth annual Capitol Hill Classic is just the thing to temper your tickle. You can register for the 10-kilometer foot race Saturday between 10 and 4 in front of Peabody Elementary, Fifth and C streets NE (Sunday registration is 7:30 a.m.). The race starts at 9 sharp in front of the school, with free refreshments at the finish line. There are also three children's races; the first will start at 10:30 in Stanton Park across C Street from the school. Entry fees are $7.50 for adults, $5 for kids under 16 (tax-deductible) and all proceeds go to the Peabody Elementary/Hobson Middle schools. Call 546-5750.