Heatmap(mat.z, 
            name = "Z-score - log2CPM", 
            col = circlize::colorRamp2(c(-2, 0, 2), c("blue", "black", "red")),
            na_col = "grey",
            cluster_rows = TRUE,
            cluster_columns = TRUE,
            right_annotation = ha,
            row_names_max_width = unit(10,"in"),
            row_title_gp = gpar(fontsize = 15),
            column_title_gp = gpar(fontsize = 20),
            column_names_gp = gpar(fontsize =10),
            row_names_gp = gpar(fontsize =3),
            show_column_names = TRUE,
            show_row_names = FALSE)

See show_row_names. Also, don't use T and F in place of TRUE and FALSE, as they are interpreted as variables and thus can be overwritten.

To follow up on Jared's suggestions - It should work.

See the minimal examples below and double-check your code against it:

set.seed(123)
test = matrix(rnorm(200), 20, 10)
test[1:10, seq(1, 10, 2)] = test[1:10, seq(1, 10, 2)] + 3
test[11:20, seq(2, 10, 2)] = test[11:20, seq(2, 10, 2)] + 2
test[15:20, seq(2, 10, 2)] = test[15:20, seq(2, 10, 2)] + 4
colnames(test) = paste("Test", 1:10, sep = "")
rownames(test) = paste("Gene", 1:20, sep = "")

genes_to_show <- c("Gene10")
mark_at = which(rownames(test) %in% genes_to_show)
ha = rowAnnotation(foo = anno_mark(at = which(rownames(test) %in% genes_to_show),
                                       labels = rownames(test)[rownames(test)%in%genes_to_show]))

Heatmap(test, name="Zscore", show_column_names = TRUE, column_names_gp = gpar(fontsize =10), right_annotation = ha, show_row_names = FALSE)

You could also do this

set.seed(123)
test = matrix(rnorm(200), 20, 10)
test[1:10, seq(1, 10, 2)] = test[1:10, seq(1, 10, 2)] + 3
test[11:20, seq(2, 10, 2)] = test[11:20, seq(2, 10, 2)] + 2
test[15:20, seq(2, 10, 2)] = test[15:20, seq(2, 10, 2)] + 4
colnames(test) = paste("Test", 1:10, sep = "")
rownames(test) = paste("Gene", 1:20, sep = "")

genes_to_show <- c("Gene10")
mark_at = which(rownames(test) %in% genes_to_show)
ha = rowAnnotation(foo = anno_mark(at = which(rownames(test) %in% genes_to_show),
                                       labels = rownames(test)[rownames(test)%in%genes_to_show]))

# Create our own row dendrogram (ComplexHeatmap orders rows by mean by default)
dend <- reorder(as.dendrogram(hclust(dist(test))), -rowMeans(test), agglo.FUN = mean)

# Find rows in dendrogram
dend_idx <- which(order.dendrogram(dend) %in% which(rownames(test) %in% genes_to_show))

# Find bottom and top of each row on heatmap (x and y axes go from 0 to 1)
btm <- 1 - (dend_idx / nrow(test))
top <- btm + (1/nrow(test))


Heatmap(test, name="Zscore", cluster_rows = dend, show_column_names = TRUE, column_names_gp = gpar(fontsize =10), right_annotation = ha, show_row_names = FALSE, border = T, border_gp = gpar(lwd=2))

box_col <- 'black'
box_width <- 2
decorate_heatmap_body("Zscore", { for (i in 1:length(genes_to_show)) {
  grid.lines(c(0, 1), c(top[i],top[i]), gp = gpar(lty = 1, lwd = box_width, col = box_col))
  grid.lines(c(0, 1), c(btm[i],btm[i]), gp = gpar(lty = 1, lwd = box_width, col = box_col))
}
})