The 'simple' way would be to draw the boxplot and the errorbars with seperate linetypes e.g. (using the palmerpenguins dataset):
library(tidyverse)
library(palmerpenguins)
penguins %>%
ggplot(aes(x = species, y = body_mass_g)) +
stat_boxplot(geom = "errorbar", width = 0.3, lty = 2) +
geom_boxplot(aes(ymin=..lower.., ymax=..upper..)) +
stat_summary(fun = mean, geom = "errorbar", aes(ymax = after_stat(y), ymin = after_stat(y), color = "mean"),
width = .75, linetype = 1, size=1) +
stat_summary(fun = median, geom = "errorbar", aes(ymax = after_stat(y), ymin = after_stat(y), color = "median"),
width = .75, linetype = 1, size=1) +
theme_minimal() +
scale_y_continuous(limits = c(0, 8000)) +
theme(legend.title=element_blank(),
legend.text = element_text(size=15, face="bold"),
legend.background = element_rect(color="black"),
legend.position =c(0.9,0.9),
legend.direction = "vertical",
axis.text.x = element_text(size=15, angle = 20, vjust=0.5),
axis.text.y = element_text(size =15),
axis.title.y = element_text(size =20),
axis.title.x=element_blank())

Created on 2022-06-01 by the reprex package (v2.0.1)
The 'more complicated' way would be to draw in the caps yourself, e.g.
library(tidyverse)
library(palmerpenguins)
penguins %>%
na.omit() %>%
group_by(species) %>%
mutate(bottom_cap = boxplot.stats(body_mass_g)$stats[1],
top_cap = boxplot.stats(body_mass_g)$stats[5]) %>%
ggplot(aes(x = species, y = body_mass_g)) +
stat_boxplot(geom = "errorbar", width = 0.3, lty = 2) +
geom_linerange(aes(xmin = as.numeric(species) - 0.2,
xmax = as.numeric(species) + 0.2,
y = top_cap)) +
geom_linerange(aes(xmin = as.numeric(species) - 0.2,
xmax = as.numeric(species) + 0.2,
y = bottom_cap)) +
geom_boxplot(aes(ymin=..lower.., ymax=..upper..)) +
stat_summary(fun = mean, geom = "errorbar", aes(ymax = after_stat(y), ymin = after_stat(y), color = "mean"),
width = .75, linetype = 1, size=1) +
stat_summary(fun = median, geom = "errorbar", aes(ymax = after_stat(y), ymin = after_stat(y), color = "median"),
width = .75, linetype = 1, size=1) +
theme_minimal() +
scale_y_continuous(limits = c(0, 8000)) +
theme(legend.title=element_blank(),
legend.text = element_text(size=15, face="bold"),
legend.background = element_rect(color="black"),
legend.position =c(0.9,0.9),
legend.direction = "vertical",
axis.text.x = element_text(size=15, angle = 20, vjust=0.5),
axis.text.y = element_text(size =15),
axis.title.y = element_text(size =20),
axis.title.x=element_blank())

Created on 2022-06-01 by the reprex package (v2.0.1)