This response is from Gary Potts, Illinois Department of Natural Resources District Wildlife Biologist.
For methods legal to hunt turtles, refer to the 2010 issue of the Illinois Fishing Information Digest http://dnr.state.il.us/fish/Digest/. On page 6 of the digest are the rules for taking turtles which follows: “Turtles may not be taken by commercial fishing devices, including hoop nets, traps or seines, or by using firearms, air guns or gas guns. Turtles may be taken only by hand, landing net or by hook and line. Locating turtles by rodding, and then taking by hand is permissible. The daily catch limit is 8 turtles (of each species) per day with a possession limit of 16 of each turtle species.” This excludes turtles protected by the Endangered Species Act. Administrative rule 880 governs the taking.
Regarding the four hooks on a line with floats: The four floats with hooks on the line was probably a trot line. A trot line is a line with hooks spaced not less than 24 inches apart. To inquire regarding the legality of the floating trot line in the Fox River please contact the Regional Fisheries Biologist or Law Enforcement Captain at the IDNR Region II office at 2050 West Stearns Road, Bartlett, IL 60103; phone: 847-608-3100. As long as the floating trot lines are legally allowed in the Fox River, the person placing them in the river was legal with a fishing license.