The issue is not specific to a GLM. It's an issue of treatment contrasts.

You should also look at the model with intercept:

set.seed(42)
y <- as.factor(sample(rep(1:2), 30, T))
x <- as.factor(sample(rep(1:2), 30, T))
z <- as.factor(sample(rep(1:2), 30, T))

fit0 <- glm(y ~ z + x, binomial)
predict(fit0, newdata=data.frame(z=factor(2), x=factor(1)))
coef(fit0)
#(Intercept)          z2          x2 
# -0.1151303   0.3228803   1.0588217 
predict(fit0, newdata=data.frame(z=factor(2), x=factor(1)))
#      1 
#0.20775 

Here the intercept represents the group x1/z1 and the other group means are calculated by adding the coefficients of z2 and/or x2.

fit1 <- glm(y ~ z + x - 1, binomial)
coef(fit1)
#        z1         z2         x2 
#-0.1151303  0.2077500  1.0588217 
predict(fit1, newdata=data.frame(z=factor(2), x=factor(1)))
#      1 
#0.20775

Here the coefficient of z1 represents the group x1/z1 which is the same as the intercept in fit0. However, the coefficient of z2 represents the group x1/z2. Note that 0.208 = -0.115 + 0.323. The x2/* group means are calculated by adding the x2 coefficient to the x1/* group means.

It should now be easy to understand why order matters here.

The issue is not specific to a GLM. It's an issue of treatment contrasts.

You should also look at the model with intercept:

set.seed(42)
y <- as.factor(sample(rep(1:2), 30, T))
x <- as.factor(sample(rep(1:2), 30, T))
z <- as.factor(sample(rep(1:2), 30, T))

fit0 <- glm(y ~ z + x, binomial)
predict(fit0, newdata=data.frame(z=factor(2), x=factor(1)))
coef(fit0)
#(Intercept)          z2          x2 
# -0.1151303   0.3228803   1.0588217 
predict(fit0, newdata=data.frame(z=factor(2), x=factor(1)))
#      1 
#0.20775 

Here the intercept represents the group x1/z1 and the other group means are calculated by adding the coefficients of z2 and/or x2.

fit1 <- glm(y ~ z + x - 1, binomial)
coef(fit1)
#        z1         z2         x2 
#-0.1151303  0.2077500  1.0588217 
predict(fit1, newdata=data.frame(z=factor(2), x=factor(1)))
#      1 
#0.20775

Here the coefficient of z1 represents the group x1/z1 which is the same as the intercept in fit0. However, the coefficient of z2 represents the group x1/z2 instead of the difference between the group means. Note that 0.208 = -0.115 + 0.323. The x2/* group means are calculated by adding the x2 coefficient to the x1/* group means.

It should now be easy to understand why order matters here.