1) The CPUID feature is detected through the ID flag 21.
If you can toggle the flag 21, the instruction is available.
The first correction, in cpu.cpp, is:

void CPU_SetFlags(Bitu word,Bitu mask) {
mask=mask|FLAG_ID; // Preserve ID flag
reg_flags=(reg_flags & ~mask)|(word & mask)|2;
cpu.direction=1-((reg_flags & FLAG_DF) >> 9);
}
WITHOUT THIS THE CPUID INSTRUCTION IS MEANINGLESS.

2) When you call CPUID with eax=0, it returns the ID string and
the maximum CPUID features. You must set the string correctly
so the emulated cpu is detected as an Intel.
When you call CPUID with eax=1, it returns the feature flags and
the cpu info; you might want to make the cpu appears to be a
Pentium so a certain game or app will run.
#define FPU_FLAG 0x0001 /* Has an FPU */
#define VME_FLAG 0x0002 /* Virtual Mode extension */

void CPU_CPUID(void) {
switch (reg_eax) {
case 0: /* Vendor ID String and maximum level */
reg_eax=1; /* Maximum level */
reg_ebx='G' | ('e' << 8 ) | ('n' << 16) | ('u'<< 24);
reg_edx='i' | ('n' << 8 ) | ('e' << 16) | ('I'<< 24);
reg_ecx='n' | ('t' << 8 ) | ('e' << 16) | ('l'<< 24);
break;
case 1: /* get processor type/family/model/stepping and feature flags */
reg_eax=0x52F; /* Intel Pentium (0x400 for an i486DX) */
reg_ebx=0; /* reserved */
reg_ecx=0; /* reserved */
reg_edx=FPU_FLAG | VME_FLAG; /* Feature flags */
break;
default:
LOG(LOG_CPU,LOG_ERROR)("Unhandled CPUID Function %x",reg_eax);
break;
}
}