Ed,
 
An article* by A.J.W. Duijvestijn (10Dec1927-21Jan1998) lists the simple imperfect squared squares of orders up to and including 19.
*: A.J.W. Duijvestijn, Electronic computation of squared rectangles, Philips Research Reports 17, 523-613, 1962 (edited by the Research Laboratory of N.V.Philips' Gloeilampenfabrieken, Eindhoven, Netherlands).
 
They include the side-91 order-18 squaring rediscovered by Robert T. Wainwright. The Bouwkamp code for this and Wainwright's other solutions to the Mrs Perkins's Quilt problem are:
Side=53 Order=16 (Compound) - (29,24)(6,5,13)(24,4,1)(5)(3,4)(7)(6,3)(18)(13);
Side=91 Order=18 (Simple) - (52,39)(15,24)(39,7,6)(4,6,5)(1,9)(8)(1,4)(7)(28)(24); 
Side=154 Order=20 (Simple) - (91,63)(24,39)(7,9,8)(63,24,4)(1,6)(5)(1,7)(4,6)(15)(52)(39).
 
The other squarings in Duijvestijn's article which beat the orders you gave for the Mrs Perkins's Quilt problem are:
Side=51 Order=16 - (22,14,15)(13,1)(16)(13,9)(9,4)(4,5)(20)(16,1)(15);
Side=52 Order=16 - (28,24)(7,9,8)(24,4)(1,6)(5)(1,7)(4,6)(15)(13);
Side=67 Order=17 - (39,28)(12,7,9)(5,2)(11)(28,8,3)(2,7,8)(5)(20)(19).
 
Duijvestijn's squarings whose orders are the same as those you gave for the Mrs Perkins's Quilt problem are:
Side=23 Order=13 - (12,11)(1,3,7)(11,2)(5)(2,5)(4,1)(3);
Side=39 Order=15 - (20,8,11)(5,3)(2,12)(7)(19,8)(5,7)(11,2)(9);
Side=39 Order=15 - (20,19)(1,3,8,7)(19,2)(5)(2,5)(12,1)(3)(8);
Side=41 Order=15 - (23,18)(7,11)(18,3,2)(1,5,3)(4)(2,1)(12)(11);
Side=48 Order=16 - (28,20)(7,5,8)(2,3)(9)(20,8)(11)(12,5)(2,9)(7);
Side=48 Order=16 - (28,20)(8,12)(20,9,7)(5,7)(2,5)(11)(3,2)(9)(8);
Side=48 Order=16 - (28,20)(11,9)(20,8)(2,7)(8,5)(5,3)(12)(2,9)(7);
Side=62 Order=17 - (33,29)(4,5,20)(29,7,1)(6)(13)(7,13)(9,4)(1,6)(5);
Side=64 Order=17 - (36,28)(9,8,11)(28,8)(3,5)(7,2)(5)(2,9)(7)(20)(16);
Side=64 Order=17 - (36,28)(9,11,8)(28,8)(3,5)(7,2)(5,9,2)(7)(20)(16);
Side=65 Order=17 - (33,32)(1,3,8,20)(32,2)(5)(13)(2,5,13)(12,1)(3)(8);
Side=65 Order=17 - (36,29)(16,13)(14,13,9)(4,9)(4,20,1)(5)(1,16)(15)(14);
Side=68 Order=17 - (25,20,23)(5,12,3)(26)(23,7)(19)(20,3)(7,19)(17,5)(12);
Side=68 Order=17 - (36,32)(4,6,7,15)(32,8)(5,1)(8)(1,4)(9)(6,21)(15);
Side=68 Order=17 - (36,32)(8,9,15)(32,4)(11,1)(10)(4,11)(17,8)(1,10)(9);
Side=69 Order=17 - (39,30)(7,12,11)(2,5)(30,11)(3,8)(8,7,2)(5)(20)(19);
Side=70 Order=17 - (38,32)(6,9,17)(32,8,4)(1,8)(5)(7,1)(6)(4,21)(17);
Side=70 Order=17 - (39,31)(5,7,19)(3,2)(1,8)(31,12)(5,3)(2,1)(20)(19);
Side=70 Order=17 - (41,29)(11,18)(2,5,4)(29,11,1)(3)(1,3)(7,2)(23)(18);
Side=80 Order=18 - (44,36)(8,28)(36,16)(9,7)(5,7,16)(2,5)(11)(3,2)(9)(8);
Side=80 Order=18 - (44,36)(11,9,16)(36,8)(2,7)(8,5)(5,3)(28)(2,9)(7)(16);
Side=80 Order=18 - (44,36)(16,7,5,8)(2,3)(9)(36,8)(11)(28,5)(2,9)(7)(16);
Side=80 Order=18 - (51,29)(14,15)(13,1)(16)(29,13,9)(9,4)(4,5)(20)(16,1)(15);
Side=82 Order=18 - (47,35)(12,23)(35,12,5,7)(3,2)(1,5,3)(4)(2,1)(24)(23);
Side=82 Order=18 - (51,31)(15,16)(9,5,1)(4,13)(31,16,4)(9)(13)(22)(15,1)(14);
Side=84 Order=18 - (44,40)(7,13,20)(40,4)(1,6)(5)(11,6,7)(5,1)(4,24)(20);
Side=84 Order=18 - (48,36)(9,7,20)(2,5)(8,3)(36,7,5)(8)(2,11)(9)(28)(20);
Side=85 Order=18 - (39,25,21)(4,17)(15,14)(1,16)(21,17,1)(15)(16)(31)(4,29)(25);
Side=85 Order=18 - (46,39)(9,11,19)(39,7)(7,2)(5,8)(5,2)(3,11)(8)(27)(19);
Side=86 Order=18 - (47,39)(7,5,8,19)(2,3)(9)(39,8)(11)(12,5)(2,28)(7)(19);
Side=86 Order=18 - (47,39)(8,12,19)(39,9,7)(5,7)(2,5)(11)(3,2)(28)(8)(19);
Side=86 Order=18 - (47,39)(11,8,20)(39,8)(3,5)(5,7,2)(7)(11,2)(9)(27)(20);
Side=86 Order=18 - (51,35)(15,20)(1,5,9)(35,13,4)(9)(4,16)(13)(22)(1,15)(14);
Side=87 Order=18 - (48,39)(7,12,20)(2,5)(39,11)(9,8)(8,3)(28)(2,7)(5)(20);
Side=87 Order=18 - (48,39)(8,11,20)(3,5)(39,7,2)(5)(2,9)(7)(12)(8,28)(20);
Side=87 Order=18 - (51,36)(14,22)(1,13)(36,16)(9,13)(4,9)(20)(5,4)(1,16)(15).
 
Geoff Morley