No, because there would have to be two recessive alleles for the foal to have a red coat.
In this cross, the stallion is homozygous dominant (EE) and can only pass on the dominant E allele. The mare is heterozygous (Ee) and can pass on either the E or the e allele. The possible genotypes for the foal from this cross are EE or Ee. Since red coat color requires the genotype ee (which means the foal must inherit the recessive allele e from both parents), it is not possible for this cross to produce a foal with a red coat.